Well playing with geogebra I have discovered (maybe not) the blue tooth's curve see the picture below :
It's well define by the function :
$$f(x)=(π x)^{2 (1-π x)}+(π (1-x))^{2 (1-π (1-x))}$$
I want to solve the integral inequality :
$$\frac{(2 + π)}{π (1 + π)}<\int_{0}^{\frac{1}{2}}f(x)dx$$
Well for that I try Taylor series (at $x=0.5$) and Bernoulli's inequality to get a polynomials without success because it's too sharp.
I don't know what other tools I can use for .
If you have an approach or an idea it's welcome !
Thanks in advance !
This would be a very difficult problem since
$$\frac{(2 + π)}{π (1 + π)} \approx 0.395167 \qquad \text{and} \qquad \int_{0}^{\frac{1}{2}}f(x)\,dx\approx 0.395543$$ and I do not think that any function approximations could be of any help with a so tiny difference.
Avoiding numerical integration, I built an interpolation function for the integrand and, itegrating, got the same result.
May be, you could try with spline functions.